Deleting a row or column of an adjacency matrix while maintaining the associated label

Question

I am currently working with a weighted adjacency matrix for a directed graph, and it contains several 0 columns and rows. With the unaltered matrix, I am able to monitor the relations between vertices with,

TableForm[Normal @ WeightedAdjacencyMatrix[graph], 
    TableHeadings -> {a = VertexList[graph], a}]

This outputs a table with the corresponding vertex list labeling the rows and columns. I want to delete the 0 rows and columns while altering the labels to reflect the change. My matrix is currently $85\times 85$, and eliminating the necessary rows and columns reduces the size to $77\times 38$. I could theoretically go through by hand and track the eliminated entries, but that sounds way too time consuming for something that I'm sure has a simple solution. Any help is appreciated.

Answer 1

graph= Graph[Range@5 , {1 -> 5, 5 -> 3, 3 -> 1}, EdgeWeight-> RandomInteger[100, 3]]

shAdj[graph_] := 
  Grid[Transpose[
    Select[Transpose[
      Select[Join[{Join[{""}, VertexList[graph]]}, 
        Transpose[Join[{VertexList[graph]}, WeightedAdjacencyMatrix[graph]]]], 
       Total@Rest@# != 0 &]], Total@Rest@# != 0 &]], 
   Alignment -> Right, Dividers -> {{2 -> Red}, {2 -> Red}}];

shAdj[graph]

Answer 2

nonzerorowsF = Function[{grph}, Pick[Range[VertexCount[grph]],
   Tr@Abs[#] != 0 & /@ WeightedAdjacencyMatrix[grph]]];
nonzerocolsF = Function[{grph}, Pick[Range[VertexCount[grph]],
   Tr@Abs[#] != 0 & /@ Transpose[WeightedAdjacencyMatrix[grph]]]];

example:

 options = Sequence[VertexStyle -> LightYellow,
  VertexSize -> 0.2,
  VertexLabels -> Placed["Name", {1/2, 1/2}],
  VertexLabelStyle -> Directive[16, Red, Bold, Italic],
  EdgeLabelStyle -> Directive[16, Blue, Bold],
  ImageSize -> 350, EdgeStyle -> Blue];


 ew = RandomReal[{-5, 5}, 4];
 g = Graph[{3, 4, 5, 1, 2, 6},
     {2 -> 3, 3 -> 1, 1 -> 2, 1 -> 4},
     EdgeWeight -> ew, options,
     EdgeLabels -> Thread[EdgeList[g] -> ew]]

rows = nonzerorowsF[g];
columns = nonzerocolsF[g];
TableForm[Normal@WeightedAdjacencyMatrix[g][[rows, columns]],
    TableHeadings -> {VertexList[g][[rows]], VertexList[g][[columns]]}]